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Solved 2024 Specimen Paper ICSE Class 10 Mathematics — Question 16

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16
Question

Question 9(ii)

The total expenses of a trip for certain number of people is ₹ 18,000. If three more people join them, then the share of each reduces by ₹ 3,000. Take x to be the original number of people, form a quadratic equation in x and solve it to find the value of x.

Answer

Let no. of people be x.

Total expense = ₹ 18,000

Expense per person = ₹ 18000x\dfrac{18000}{x}

Given,

If three more people join them, then the share of each reduces by ₹ 3,000.

No, of people now = x + 3

Expense per person = ₹ 18000x+3\dfrac{18000}{x + 3}

According to question,

⇒18000x−18000x+3=3000⇒18000(x+3)−18000xx(x+3)=3000⇒18000x+54000−18000xx2+3x=3000⇒54000x2+3x=3000⇒x2+3x=540003000⇒x2+3x=18⇒x2+3x−18=0⇒x2+6x−3x−18=0⇒x(x+6)−3(x+6)=0⇒(x−3)(x+6)=0⇒x−3=0 or x+6=0⇒x=3 or x=−6.\Rightarrow \dfrac{18000}{x} - \dfrac{18000}{x + 3} = 3000 \\[1em] \Rightarrow \dfrac{18000(x + 3) - 18000x}{x(x + 3)} = 3000 \\[1em] \Rightarrow \dfrac{18000x + 54000 - 18000x}{x^2 + 3x} = 3000 \\[1em] \Rightarrow \dfrac{54000}{x^2 + 3x} = 3000 \\[1em] \Rightarrow x^2 + 3x = \dfrac{54000}{3000} \\[1em] \Rightarrow x^2 + 3x = 18 \\[1em] \Rightarrow x^2 + 3x - 18 = 0 \\[1em] \Rightarrow x^2 + 6x - 3x - 18 = 0 \\[1em] \Rightarrow x(x + 6) - 3(x + 6) = 0 \\[1em] \Rightarrow (x - 3)(x + 6) = 0 \\[1em] \Rightarrow x - 3 = 0 \text{ or } x + 6 = 0 \\[1em] \Rightarrow x = 3 \text{ or } x = -6.

Since, no. of people cannot be negative.

∴ x = 3.

Hence, original number of people = 3.